Geometry of the eigencurve at critical Eisenstein series of weight 2

Majumdar, Dipramit

doi:10.5802/jtnb.898

Majumdar, Dipramit ¹

¹ Indian Statistical Institute, Bangalore Centre 8th Mile, Mysore Road, RVCE Post, Bangalore, India 560059

Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 1, pp. 183-197.

Résumé
Abstract

Dans cet article, nous montrons que la série d’Eisenstein critique de poids 2, $E_{2}^{c r i t_{p}}$ , définit un point lisse dans la courbe de Hecke $ℂ (l)$ , où $l$ est un nombre premier différent de $p$ . Nous montrons également que $E_{2}^{c r i t_{p}, o r d_{l}}$ définit un point lisse dans la courbe de Hecke pleine $ℂ^{f u l l} (l)$ et que le point défini par $E_{2}^{c r i t_{p}, o r d_{l_{1}}, o r d_{l_{2}}}$ est non lisse dans la courbe de Hecke pleine $ℂ^{f u l l} (l_{1} l_{2})$ . En outre, nous montrons que $ℂ (l)$ est étale sur l’espace des poids au point défini par $E_{2}^{c r i t_{p}}$ . En conséquence, nous montrons que la conjecture d’abaissement du niveau de Paulin n’est pas valide pour $E_{2}^{c r i t_{p}, o r d_{l}}$ .

In this paper we show that the critical Eisenstein series of weight 2, $E_{2}^{c r i t_{p}}$ , defines a smooth point in the eigencurve $ℂ (l)$ , where $l$ is a prime different from $p$ . We also show that $E_{2}^{c r i t_{p}, o r d_{l}}$ defines a smooth point in the full eigencurve $ℂ^{f u l l} (l)$ and $E_{2}^{c r i t_{p}, o r d_{l_{1}}, o r d_{l_{2}}}$ defines a non-smooth point in the full eigencurve $ℂ^{f u l l} (l_{1} l_{2})$ . Further, we show that $ℂ (l)$ is étale over the weight space at the point defined by $E_{2}^{c r i t_{p}}$ . As a consequence, we show that level lowering conjecture of Paulin fails to hold at $E_{2}^{c r i t_{p}, o r d_{l}}$ .

MR Zbl | 2 citations dans Numdam

DOI : 10.5802/jtnb.898

Classification : 11F85, 11F11, 11F80

Affiliations des auteurs :

Majumdar, Dipramit ¹

¹ Indian Statistical Institute, Bangalore Centre 8th Mile, Mysore Road, RVCE Post, Bangalore, India 560059

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     title = {Geometry of the eigencurve at critical {Eisenstein} series of weight 2},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {183--197},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
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     year = {2015},
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Majumdar, Dipramit. Geometry of the eigencurve at critical Eisenstein series of weight 2. Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 1, pp. 183-197. doi : 10.5802/jtnb.898. https://www.numdam.org/articles/10.5802/jtnb.898/

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